Abstract Linear Algebra Math 350 April 29, 2015 Contents 1 An introduction to vector spaces 2 1.1 Basic definitions & preliminaries . .4 1.2 Basic algebraic properties of vector spaces . .6 1.3 Subspaces . .7 2 Dimension 10 2.1 Linear combination . 10 2.2 Bases . 13 2.3 Dimension . 14 2.4 Zorn's lemma & the basis extension theorem . 16 3 Linear transformations 18 3.1 Definition & examples . 18 3.2 Rank-nullity theorem . 21 3.3 Vector space isomorphisims . 26 3.4 The matrix of a linear transformation . 28 4 Complex operators 34 4.1 Operators & polynomials . 34 4.2 Eigenvectors & eigenvalues . 36 4.3 Direct sums . 45 4.4 Generalized eigenvectors . 49 4.5 The characteristic polynomial . 54 4.6 Jordan basis theorem . 56 1 Chapter 1 An introduction to vector spaces Abstract linear algebra is one of the pillars of modern mathematics. Its theory is used in every branch of mathematics and its applications can be found all around our everyday life. Without linear algebra, modern conveniences such as the Google search algorithm, iPhones, and microprocessors would not exist. But what is abstract linear algebra? It is the study of vectors and functions on vectors from an abstract perspective. To explain what we mean by an abstract perspective, let us jump in and review our familiar notion of vectors. Recall that a vector of length n is a n × 1 array 2 3 a1 6 a2 7 6 7 ; 6 . 7 4 . 5 an where ai are real numbers, i.e., ai 2 R. It is also customary to define 82 3 9 a1 > > <>6 a2 7 => n = 6 7 j a 2 ; R 6 . 7 i R >4 . 5 > > > : an ; which we can think of as the set where all the vectors of length n live. Some of the usefulness of vectors stems from our ability to draw them (at least those in 2 3 R or R ). Recall that this is done as follows: 2 z y 2 a 3 a b b b 4 5 x c c a a y b x Basic algebraic operations on vectors correspond nicely with our picture of vectors. In particular, if we scale a vector v by a number s then in the picture we either stretch or shrink our arrow. y sb a s · b b a x sa The other familiar thing we can do with vectors is add them. This corre- sponds to placing the vectors \head-to-tail" as shown in the following picture. y c b d a c + b + d b d x a a + c In summary, our familiar notion of vectors can be captured by the following n description. Vectors of length n live in the set R that is equipped with two n operations. The first operation takes any pair of vectors u; v 2 R and gives n us a new vector u + v 2 R . The second operation takes any pair a 2 R and n n v 2 R and gives us a new vector a · v 2 R . With this summary in mind we now give a definition which generalizes this familiar notion of a vector. It will be very helpful to read the following in parallel with the above summary. 3 1.1 Basic definitions & preliminaries Throughout we let F represent either the rational numbers Q, the real numbers R or the complex numbers C. Definition. A vector space over F is a set V along with two operation. The first operation is called addition, denoted +, which assigns to each pair u; v 2 V an element u + v 2 V . The second operation is called scalar multiplication which assigns to each pair a 2 F and v 2 V an element av 2 V . Moreover, we insist that the following properties hold, where u; v; w 2 V and a; b 2 F: • Associativity u + (v + w) = (u + v) + w and a(bv) = (ab)v: • Commutativity of + u + v = v + u: • Distributivity a(u + v) = au + av and (a + b)v = av + bv • Multiplicative Identity The number 1 2 F is such that 1v = v for all v 2 V: • Additive Identity & Inverses There exists an element 0 2 V , called an additive identity or a zero, with the property that 0 + v = v for all v 2 V: Moreover, for every v 2 V there exists some u 2 V , called an inverse of v, such that u + v = 0. It is common to refer to the elements of V as vectors and the elements of F as scalars. Additionally, if V is a vector space over R we call it a real vector space or an R-vector space. Likewise, a vector space over C is called a complex vector space or a C-vector space. Although this definition is intimidating at first, you are more familiar with these ideas than you might think. In fact, you have been using vector spaces in your previous math courses without even knowing it! The following examples aim to convince you of this. Examples. 4 n 1. R is a vector space over R under the usual vector addition and scalar multiplication as discussed in the introduction. n 2. C , the set of column vectors of length n whose entries are complex num- bers, is a a vector space over C. n 3. C is also a vector space over R where addition is standard vector addition and scalar multiplication is again the standard operation but in this case we limit our scalars to real numbers only. This is NOT the same vector space as in the previous example; in fact, it is as different as a line is to a plane! 4. Let P(F) be the set of all polynomials with coefficients in F. That is n P(F) = fa0 + a1x + ··· + anx j n ≥ 0; a0; : : : ; an 2 Fg : Then P(F) is a vector space over F. In this case our \vectors" are poly- nomials where addition is the standard addition on polynomials. For ex- ample, if v = 1 + x + 3x2 and u = x + 7x2 + x5, then u + v = (1 + x + 3x2) + (x + 7x2 + x5) = 1 + 2x + 10x2 + x5: Scalar multiplication is defined just as you might think. If v = a0 + a1x + n ··· + anx , then n s · v = sa0 + sa1x + ··· + sanx : 5. Let C(R) be the set of continuous functions f : R ! R : Then C(R) is a vector space over R where addition and scalar multiplica- tion is given as follows. For any functions f; g 2 C(R) we define (f + g)(x) = f(x) + g(x): Likewise, for scalar multiplication we define (s · f)(x) = sf(x): The reader should check that these definitions satisfy the axioms for a vector space. 6. Let F be the set of all functions f : R ! R. Then the set F is a vector space over R where addition and scalar multiplication are as given in Example 5. You might be curious why we use the term \over" when saying that a vector space V is over F. The reason for this is due to a useful way to visualize abstract vector spaces. In particular, we can draw the following picture 5 V F where our set V is sitting over our scalars F. 1.2 Basic algebraic properties of vector spaces n There are certain algebraic properties that we take for granted in R . For example, the zero vector 2 0 3 6 . 7 n 4 . 5 2 R 0 n n is the unique additive identity in R . Likewise, in R we do not even think about the fact that −v is the (unique) additive inverse of v. These algebraic properties are so fundamental that we certainly would like our general vector spaces to have these same properties as well. As the next several lemmas show, this is happily the case. Assume throughout this section that V is a vector space over F. Lemma 1.1. V has a unique additive identity. Proof. Assume 0 and 00 are both additive identities in V . To show V has a unique additive identity we show that 0 = 00. Playing these two identities off each other we see that 00 = 0 + 00 = 0; where the first equality follows as 0 is an identity and the second follows since 00 is also an identity. An immediate corollary of this lemma is that now we can talk about the additive identity or the zero of a vector space. To distinguish between zero, the number in F and the zero the additive identity in V we will often denote the latter as 0V . Lemma 1.2. Every element v 2 V has a unique additive inverse denoted −v. Proof. Fix v 2 V . As in the proof of the previous lemma, it will suffice to show that if u and u0 are both additive inverses of v, then u = u0. Now consider 0 0 0 0 u = 0V + u = (u + v) + u = u + (v + u ) = u + 0V = u; where associativity gives us the third equality. 6 Lemma 1.3 (Cancellation Lemma). If u; v; w are vectors in V such that u + w = v + w; (*) then u = v Proof. To show this, add −w to both sides of (*) to obtain (u + w) + −w = (v + w) + −w. By associativity, u + (w + −w) = v + (w + −w) u + 0V = v + 0V u = v: Lemma 1.4. For any a 2 F and v 2 V , we have 0 · v = 0V and a · 0V = 0V : Proof. The proof of this is similar to the Cancellation Lemma.